doi: 10.3934/dcdsb.2021263
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo

1. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

2. 

Zhejiang Institute, China University of Geosciences, Hangzhou, Zhejiang 311305, China

3. 

Beijing University of Technology, Beijing, 100124, China

* Corresponding author: weizhouchao@163.com

Received  April 2021 Revised  August 2021 Early access October 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (Grant No.11772306, 12172340), Zhejiang Provincial Natural Science Foundation of China under Grant (No.LY20A020001), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (CUGGC05). The last author is supported by National Natural Science Foundation of China (Grant No. 11832002)

In this paper, we make a thorough inquiry about the Jacobi stability of 5D self-exciting homopolar disc dynamo system on the basis of differential geometric methods namely Kosambi-Cartan-Chern theory. The Jacobi stability of the equilibria under specific parameter values are discussed through the characteristic value of the matrix of second KCC invariants. Periodic orbit is proved to be Jacobi unstable. Then we make use of the deviation vector to analyze the trajectories behaviors in the neighborhood of the equilibria. Instability exponent is applicable for predicting the onset of chaos quantitatively. In addition, we also consider impulsive control problem and suppress hidden attractor effectively in the 5D self-exciting homopolar disc dynamo.

Citation: Zhouchao Wei, Fanrui Wang, Huijuan Li, Wei Zhang. Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021263
References:
[1]

H. Abolghasem, Liapunov stability versus Jacobi stability, Journal of Dynamical Systems and Geometric Theories, 10 (2012), 13-32.  doi: 10.1080/1726037X.2012.10698604.  Google Scholar

[2]

H. Abolghasem, Jacobi stability of circular orbits in a central force, J. Dyn. Syst. Geom. Theor., 10 (2012), 197-214.  doi: 10.1080/1726037X.2012.10698621.  Google Scholar

[3]

P. L. Antonelli and I. Bucataru, KCC theory of a system of second order differential equations, Handbook of Finsler Geometry, 1 (2003), 83-174.   Google Scholar

[4]

J. BaoD. ChenY. Liu and H. Deng, Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6053-6069.  doi: 10.3934/dcdsb.2019130.  Google Scholar

[5]

J. Bao and Y. Liu, Multistability and bifurcations in a 5D segmented disc dynamo with a curve of equilibria, Adv. Difference Equ., 2019 (2019), 345-360.  doi: 10.1186/s13662-019-2284-0.  Google Scholar

[6]

B. C. BaoJ. LuoH. BaoC. ChenH. Wu and Q. Xu, A simple nonautonomous hidden chaotic system with a switchable stable node-focus, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950168.  doi: 10.1142/S0218127419501682.  Google Scholar

[7]

I. A. Bilenko, Coronal holes and the solar polar field reversal, Astronomy and Astrophysics, 396 (2002), 657-666.  doi: 10.1051/0004-6361:20021412.  Google Scholar

[8]

C. G. BoehmerT. Harko and S. V. Sabau, Jacobi stability analysis of dynamical systems-applications in gravitation and cosmology, Adv. Theor. Math. Phys., 16 (2012), 1145-1196.  doi: 10.4310/ATMP.2012.v16.n4.a2.  Google Scholar

[9]

E. Bullard, The stability of a homopolar dynamo, Proceedings of the Cambridge Philosophical Society, 51 (1955), 744-760.   Google Scholar

[10]

C. FengQ. Huang and Y. Liu, Jacobi analysis for an unusual 3D autonomous system, Int. J. Geom. Methods Mod. Phys., 17 (2020), 2050062.  doi: 10.1142/S0219887820500620.  Google Scholar

[11]

Y. FengZ. WeiU. E. KocamazA. Akgül and I. Moroz, Synchronization and electronic circuit application of hidden hyperchaos in a four-dimensional self-exciting homopolar disc dynamo without equilibria, Complexity, 2017 (2017), 1-11.  doi: 10.1155/2017/7101927.  Google Scholar

[12]

M. K. Gupta and C. K. Yadav, Rabinovich-Fabrikant system in view point of KCC theory in Finsler geometry, Journal of Interdisciplinary Mathematics, 22 (2019), 219-241.  doi: 10.1080/09720502.2019.1614249.  Google Scholar

[13]

T. HarkoC. Y. HoC. S. Leung and S. Yip, Jacobi stability analysis of the Lorenz system, Adv. Theor. Math. Phys., 12 (2015), 1550081.  doi: 10.1142/S0219887815500814.  Google Scholar

[14]

T. HarkoP. Pantaragphong and S. V. Sabau, Kosambi-Cartan-Chern (KCC) theory for higher order dynamical systems, Int. J. Geom. Methods Mod. Phys., 13 (2016), 1650014.  doi: 10.1142/S0219887816500146.  Google Scholar

[15]

Q. HuangA. Liu and Y. Liu, Jacobi stability analysis of the Chen system, International Journal of Bifurcation and Chaos, 29 (2019), 1950139.  doi: 10.1142/S0218127419501396.  Google Scholar

[16] J. A. Jacobs, Reversals of the Earth's Magnetic Field, Cambridge: University Press, 1994.  doi: 10.1111/j.1365-246X.1995.tb06453.x.  Google Scholar
[17]

E. Knobloch, Chaos in the segmented disc dynamo, Phys. Lett. A, 82 (1981), 439-440.  doi: 10.1016/0375-9601(81)90274-7.  Google Scholar

[18]

G. H. Kom, L. P. N. Nguenjou, C. Ainamon and S. T. Kingni, Theoretical and experimental investigations of a jerk circuit with two parallel diodes, Chaos Theory and Applications, 2 (2020), 52–57, https: //www.researchgate.net/publication/342946507. Google Scholar

[19]

G. A. LeonovN. V. Kuznetsov and V. I. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A, 375 (2011), 2230-2233.  doi: 10.1016/j.physleta.2011.04.037.  Google Scholar

[20]

H. J. Li and A. P. Liu, Asymptotic stability analysis via indefinite Lyapunov functions and design of nonlinear impulsive control systems, Nonlinear Anal. Hybrid Syst., 38 (2020), 100936.  doi: 10.1016/j.nahs.2020.100936.  Google Scholar

[21]

H. J. LiA. P. Liu and L. L. Zhang, Input-to-state stability of time-varying nonlinear discrete-time systems via indefinite difference Lyapunov functions, ISA Transactions, 77 (2018), 71-76.  doi: 10.1016/j.isatra.2018.03.022.  Google Scholar

[22]

Y. Liu, J. Li, Z. Wei and I. Moroz, Bifurcation analysis and integrability in the segmented disc dynamo with mechanical friction, Adv. Difference Equ., 2018 (2018), 22pp. doi: 10.1186/s13662-018-1659-y.  Google Scholar

[23]

H. K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophysical and Astrophysical Fluid Dynamics, 14 (1979), 147-166.  doi: 10.1080/03091927908244536.  Google Scholar

[24]

C. Y. NingY. HeM. WuQ. P. Liu and J. H. She, Input-to-state stability of nonlinear systems based on an indefinite Lyapunov function, Systems Control Lett., 61 (2012), 1254-1259.  doi: 10.1016/j.sysconle.2012.08.009.  Google Scholar

[25]

Z. T. NjitackeT. F. FozinL. K. KamdjeuG. D. LeutchoE. M. Kengne and J. Kengne, Multistability and its Annihilation in the Chua's Oscillator with Piecewise-Linear Nonlinearity, Chaos Theory and Applications, 2 (2020), 77-89.   Google Scholar

[26]

M. Y. Reshetnyak and V. E. Pavlov, Evolution of the dipole geomagnetic field. Observations and models, Geomagnetism and Aeronomy, 56 (2016), 110-124.  doi: 10.1134/S0016793215060122.  Google Scholar

[27]

T. Rikitake, Oscillations of a system of disk dynamos, Proc. Cambridge Philos. Soc., 54 (1958), 89-105.  doi: 10.1017/S0305004100033223.  Google Scholar

[28]

S. V. Sabau, Some remarks on Jacobi stability, Nonlinear Analysis, 63 (2005), 143-153.   Google Scholar

[29]

S. V. Sabau, Systems biology and deviation curvature tensor, Nonlinear Anal. Real World Appl., 6 (2005), 563-587.  doi: 10.1016/j.nonrwa.2004.12.012.  Google Scholar

[30]

M. ShahzadV. T. PhamM. A. AhmadS. Jafari and F. Hadaeghi, Synchronization and circuit design of a chaotic system with coexisting hidden attractors, The European Physical Journal Special Topics, 224 (2015), 1637-1652.  doi: 10.1140/epjst/e2015-02485-8.  Google Scholar

[31]

S. N. Shore, Magnetic fields in astrophysics, Encyclopedia of Physical Science and Technology (Third Edition), (2003), 903–918. doi: 10.1016/B0-12-227410-5/00392-6.  Google Scholar

[32]

A. S. K. TsafackR. KengneA. CheukemJ. R. M. Pone and G. Kenne, Chaos control using self-feedback delay controller and electronic implementation in IFOC of 3-phase induction motor, Chaos Theory and Applications, 1 (2020), 40-48.   Google Scholar

[33]

Z. WeiA. AkgülU. E. KocamazI. Moroz and W. Zhang, Control, electronic circuit application and fractional-order analysis of hidden chaotic attractors in the self-exciting homopolar disc dynamo, Chaos, Solitons Fractals, 111 (2018), 157-168.  doi: 10.1016/j.chaos.2018.04.020.  Google Scholar

[34]

Z. WeiI. MorozJ. C. SprottA. Akgul and W. Zhang, Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27 (2017), 033101.  doi: 10.1063/1.4977417.  Google Scholar

[35]

Z. WeiI. MorozJ. C. SprottZ. Wang and W. Zhang, Detecting hidden chaotic regions and complex dynamics in the self-exciting homopolar disc dynamo, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1730008.  doi: 10.1142/S0218127417300087.  Google Scholar

[36]

K. Yamasaki and T. Yajima, Lotka-Volterra system and KCC theory: Differential geometric structure of competitions and predations, Nonlinear Anal. Real World Appl., 14 (2013), 1845-1853.  doi: 10.1016/j.nonrwa.2012.11.015.  Google Scholar

[37]

K. Yamasaki and T. Yajima, Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory, J. Dyn. Syst. Geom. Theor., 14 (2016), 137-153.  doi: 10.1080/1726037X.2016.1250500.  Google Scholar

show all references

References:
[1]

H. Abolghasem, Liapunov stability versus Jacobi stability, Journal of Dynamical Systems and Geometric Theories, 10 (2012), 13-32.  doi: 10.1080/1726037X.2012.10698604.  Google Scholar

[2]

H. Abolghasem, Jacobi stability of circular orbits in a central force, J. Dyn. Syst. Geom. Theor., 10 (2012), 197-214.  doi: 10.1080/1726037X.2012.10698621.  Google Scholar

[3]

P. L. Antonelli and I. Bucataru, KCC theory of a system of second order differential equations, Handbook of Finsler Geometry, 1 (2003), 83-174.   Google Scholar

[4]

J. BaoD. ChenY. Liu and H. Deng, Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6053-6069.  doi: 10.3934/dcdsb.2019130.  Google Scholar

[5]

J. Bao and Y. Liu, Multistability and bifurcations in a 5D segmented disc dynamo with a curve of equilibria, Adv. Difference Equ., 2019 (2019), 345-360.  doi: 10.1186/s13662-019-2284-0.  Google Scholar

[6]

B. C. BaoJ. LuoH. BaoC. ChenH. Wu and Q. Xu, A simple nonautonomous hidden chaotic system with a switchable stable node-focus, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950168.  doi: 10.1142/S0218127419501682.  Google Scholar

[7]

I. A. Bilenko, Coronal holes and the solar polar field reversal, Astronomy and Astrophysics, 396 (2002), 657-666.  doi: 10.1051/0004-6361:20021412.  Google Scholar

[8]

C. G. BoehmerT. Harko and S. V. Sabau, Jacobi stability analysis of dynamical systems-applications in gravitation and cosmology, Adv. Theor. Math. Phys., 16 (2012), 1145-1196.  doi: 10.4310/ATMP.2012.v16.n4.a2.  Google Scholar

[9]

E. Bullard, The stability of a homopolar dynamo, Proceedings of the Cambridge Philosophical Society, 51 (1955), 744-760.   Google Scholar

[10]

C. FengQ. Huang and Y. Liu, Jacobi analysis for an unusual 3D autonomous system, Int. J. Geom. Methods Mod. Phys., 17 (2020), 2050062.  doi: 10.1142/S0219887820500620.  Google Scholar

[11]

Y. FengZ. WeiU. E. KocamazA. Akgül and I. Moroz, Synchronization and electronic circuit application of hidden hyperchaos in a four-dimensional self-exciting homopolar disc dynamo without equilibria, Complexity, 2017 (2017), 1-11.  doi: 10.1155/2017/7101927.  Google Scholar

[12]

M. K. Gupta and C. K. Yadav, Rabinovich-Fabrikant system in view point of KCC theory in Finsler geometry, Journal of Interdisciplinary Mathematics, 22 (2019), 219-241.  doi: 10.1080/09720502.2019.1614249.  Google Scholar

[13]

T. HarkoC. Y. HoC. S. Leung and S. Yip, Jacobi stability analysis of the Lorenz system, Adv. Theor. Math. Phys., 12 (2015), 1550081.  doi: 10.1142/S0219887815500814.  Google Scholar

[14]

T. HarkoP. Pantaragphong and S. V. Sabau, Kosambi-Cartan-Chern (KCC) theory for higher order dynamical systems, Int. J. Geom. Methods Mod. Phys., 13 (2016), 1650014.  doi: 10.1142/S0219887816500146.  Google Scholar

[15]

Q. HuangA. Liu and Y. Liu, Jacobi stability analysis of the Chen system, International Journal of Bifurcation and Chaos, 29 (2019), 1950139.  doi: 10.1142/S0218127419501396.  Google Scholar

[16] J. A. Jacobs, Reversals of the Earth's Magnetic Field, Cambridge: University Press, 1994.  doi: 10.1111/j.1365-246X.1995.tb06453.x.  Google Scholar
[17]

E. Knobloch, Chaos in the segmented disc dynamo, Phys. Lett. A, 82 (1981), 439-440.  doi: 10.1016/0375-9601(81)90274-7.  Google Scholar

[18]

G. H. Kom, L. P. N. Nguenjou, C. Ainamon and S. T. Kingni, Theoretical and experimental investigations of a jerk circuit with two parallel diodes, Chaos Theory and Applications, 2 (2020), 52–57, https: //www.researchgate.net/publication/342946507. Google Scholar

[19]

G. A. LeonovN. V. Kuznetsov and V. I. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A, 375 (2011), 2230-2233.  doi: 10.1016/j.physleta.2011.04.037.  Google Scholar

[20]

H. J. Li and A. P. Liu, Asymptotic stability analysis via indefinite Lyapunov functions and design of nonlinear impulsive control systems, Nonlinear Anal. Hybrid Syst., 38 (2020), 100936.  doi: 10.1016/j.nahs.2020.100936.  Google Scholar

[21]

H. J. LiA. P. Liu and L. L. Zhang, Input-to-state stability of time-varying nonlinear discrete-time systems via indefinite difference Lyapunov functions, ISA Transactions, 77 (2018), 71-76.  doi: 10.1016/j.isatra.2018.03.022.  Google Scholar

[22]

Y. Liu, J. Li, Z. Wei and I. Moroz, Bifurcation analysis and integrability in the segmented disc dynamo with mechanical friction, Adv. Difference Equ., 2018 (2018), 22pp. doi: 10.1186/s13662-018-1659-y.  Google Scholar

[23]

H. K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophysical and Astrophysical Fluid Dynamics, 14 (1979), 147-166.  doi: 10.1080/03091927908244536.  Google Scholar

[24]

C. Y. NingY. HeM. WuQ. P. Liu and J. H. She, Input-to-state stability of nonlinear systems based on an indefinite Lyapunov function, Systems Control Lett., 61 (2012), 1254-1259.  doi: 10.1016/j.sysconle.2012.08.009.  Google Scholar

[25]

Z. T. NjitackeT. F. FozinL. K. KamdjeuG. D. LeutchoE. M. Kengne and J. Kengne, Multistability and its Annihilation in the Chua's Oscillator with Piecewise-Linear Nonlinearity, Chaos Theory and Applications, 2 (2020), 77-89.   Google Scholar

[26]

M. Y. Reshetnyak and V. E. Pavlov, Evolution of the dipole geomagnetic field. Observations and models, Geomagnetism and Aeronomy, 56 (2016), 110-124.  doi: 10.1134/S0016793215060122.  Google Scholar

[27]

T. Rikitake, Oscillations of a system of disk dynamos, Proc. Cambridge Philos. Soc., 54 (1958), 89-105.  doi: 10.1017/S0305004100033223.  Google Scholar

[28]

S. V. Sabau, Some remarks on Jacobi stability, Nonlinear Analysis, 63 (2005), 143-153.   Google Scholar

[29]

S. V. Sabau, Systems biology and deviation curvature tensor, Nonlinear Anal. Real World Appl., 6 (2005), 563-587.  doi: 10.1016/j.nonrwa.2004.12.012.  Google Scholar

[30]

M. ShahzadV. T. PhamM. A. AhmadS. Jafari and F. Hadaeghi, Synchronization and circuit design of a chaotic system with coexisting hidden attractors, The European Physical Journal Special Topics, 224 (2015), 1637-1652.  doi: 10.1140/epjst/e2015-02485-8.  Google Scholar

[31]

S. N. Shore, Magnetic fields in astrophysics, Encyclopedia of Physical Science and Technology (Third Edition), (2003), 903–918. doi: 10.1016/B0-12-227410-5/00392-6.  Google Scholar

[32]

A. S. K. TsafackR. KengneA. CheukemJ. R. M. Pone and G. Kenne, Chaos control using self-feedback delay controller and electronic implementation in IFOC of 3-phase induction motor, Chaos Theory and Applications, 1 (2020), 40-48.   Google Scholar

[33]

Z. WeiA. AkgülU. E. KocamazI. Moroz and W. Zhang, Control, electronic circuit application and fractional-order analysis of hidden chaotic attractors in the self-exciting homopolar disc dynamo, Chaos, Solitons Fractals, 111 (2018), 157-168.  doi: 10.1016/j.chaos.2018.04.020.  Google Scholar

[34]

Z. WeiI. MorozJ. C. SprottA. Akgul and W. Zhang, Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27 (2017), 033101.  doi: 10.1063/1.4977417.  Google Scholar

[35]

Z. WeiI. MorozJ. C. SprottZ. Wang and W. Zhang, Detecting hidden chaotic regions and complex dynamics in the self-exciting homopolar disc dynamo, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1730008.  doi: 10.1142/S0218127417300087.  Google Scholar

[36]

K. Yamasaki and T. Yajima, Lotka-Volterra system and KCC theory: Differential geometric structure of competitions and predations, Nonlinear Anal. Real World Appl., 14 (2013), 1845-1853.  doi: 10.1016/j.nonrwa.2012.11.015.  Google Scholar

[37]

K. Yamasaki and T. Yajima, Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory, J. Dyn. Syst. Geom. Theor., 14 (2016), 137-153.  doi: 10.1080/1726037X.2016.1250500.  Google Scholar

Figure 1.  The phase portraits of hyperchaotic dynamo system with the parameter values for $ (m,g,r,k_{1},k_{2}) = (0.04,140.6,7,34,12) $: (a) $ x $-$ y $ plane; (b) time series of $ x(t) $
Figure 2.  Phase portraits of periodic orbit: (a) $ x $-$ y $ plane; (b) $ x $-$ z $ plane; (c) $ x $-$ u $ plane; (d) $ x $-$ v $ plane; (e) time series of $ x(t) $
Figure 3.  Part of time variation figures for four judgment conditions with $ (m,g,r,k_{1},k_{2}) = (0.04,140.6,3.5,34,12) $
Figure 4.  Deviation vector near $ E_{1} $ under the initial condition $ (\xi_{1},\xi_{2},\xi_{3},\xi_{4}) = (0,0,0,0) $ and different values for $ (\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) $: (a) $ (\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-8},10^{-8},10^{-8},10^{-8}) $; (b) $ (\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-5},10^{-5},10^{-5},10^{-5}) $
Figure 5.  Deviation vector near $ E_{2} $ under the initial condition $ (\xi_{1},\xi_{2},\xi_{3},\xi_{4}) = (0,0,0,0) $ and different values for $ (\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) $: (a) $ (\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-8},10^{-8},10^{-8},10^{-8}) $; (b) $ (\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-5},10^{-5},10^{-5},10^{-5}) $
Figure 6.  Time-variation of instability exponent near the equilibria under the initial condition $ (\xi_{1},\xi_{2},\xi_{3},\xi_{4}) = (0,0,0,0) $ and $ (\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-8},10^{-8},10^{-8},10^{-8}) $ (left), $ (\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-5},10^{-5},10^{-5},10^{-5}) $ (right)
Figure 7.  (a) Time series of system (31) with hidden hyperchaotic attractor; (b) The trajectory of the indefinite Lyapunov function $ V $ for system (32) with and the resetting time $ t_{k} = 0.01 k,\,\, k = 1,2,\cdots $. The initial condition we choose is $ (-5.6692,0.1119, -20.3280, -36.3177,-114.4281) $
Figure 8.  (a) Time series of system (31) with hidden hyperchaotic attractor; (b) The trajectory of the indefinite Lyapunov function $ V $ for system (32) and the resetting time $ t_{k} = 0.01 k,\,\, k = 1,2,\cdots. $ The initial condition we choose is $ (8,2, 2, 1,1) $
[1]

Jianghong Bao. Complex dynamics in the segmented disc dynamo. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3301-3314. doi: 10.3934/dcdsb.2016098

[2]

Jianghong Bao, Dandan Chen, Yongjian Liu, Hongbo Deng. Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6053-6069. doi: 10.3934/dcdsb.2019130

[3]

Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021076

[4]

Gang Tian. Bott-Chern forms and geometric stability. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 211-220. doi: 10.3934/dcds.2000.6.211

[5]

C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial & Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435

[6]

Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121

[7]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[8]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1091-1116. doi: 10.3934/dcdsb.2004.4.1091

[9]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth. Communications on Pure & Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527

[10]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313

[11]

Yingjing Shi, Rui Li, Honglei Xu. Control augmentation design of UAVs based on deviation modification of aerodynamic focus. Journal of Industrial & Management Optimization, 2015, 11 (1) : 231-240. doi: 10.3934/jimo.2015.11.231

[12]

Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275

[13]

Aram Arutyunov, Dmitry Karamzin, Fernando L. Pereira. On a generalization of the impulsive control concept: Controlling system jumps. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 403-415. doi: 10.3934/dcds.2011.29.403

[14]

Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1

[15]

Xueyan Yang, Xiaodi Li, Qiang Xi, Peiyong Duan. Review of stability and stabilization for impulsive delayed systems. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1495-1515. doi: 10.3934/mbe.2018069

[16]

Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317

[17]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683

[18]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421

[19]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[20]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]